Peso Problems in the Estimation of the C-CAPM (a,c,e) (b,c) (d) Juan Carlos Parra-Alvarez , Olaf Posch∗ and Andreas Schrimpf (a)Aarhus University, (b)Universit¨at Hamburg, (c)CREATES (d)Bank for International Settlements, CEPR, (e)Danish Finance Institute March 2021 Abstract This paper shows that the consumption-based capital asset pricing model (C-CAPM) with low-probability disaster risk rationalizes pricing errors. We find that implausible estimates of risk aversion and time preference are not puzzling if market participants expect a future catastrophic change in fundamentals, which just happens not to occur in the sample (a ‘peso problem’). A bias in structural parameter estimates emerges as a result of pricing errors in quiet times. While the bias essentially removes the pricing error in the simple models when risk-free rates are constant, time-variation may also generate large and persistent estimated pricing errors in simulated data. We also show analytically how the problem of biased estimates can be avoided in empirical research by resolving the misspecification in moment conditions. JEL classification: E21, G12, O41 Keywords: Rare events, Asset pricing errors, C-CAPM ∗Corresponding author: Olaf Posch ([email protected], Phone: +49-40-42838-4630, Address: Universit¨at Hamburg, Department of Economics, Von-Melle-Park 5, 20146 Hamburg, Germany). We thank Jessica Wachter, Joachim Grammig, and conference participants at the CEF, VfS and EEA/ESEM meetings for valuable comments. The authors appreciate financial support from the Center for Research in Econo- metric Analysis of Time Series, CREATES, funded by The Danish National Research Foundation. The views expressed in this paper are those of the authors and do not necessarily reflect those of the Bank for International Settlements. 1 Introduction It is a widespread perception that the workhorse of financial economics – the consumption- based capital asset pricing model (C-CAPM) of Rubinstein (1976), Lucas (1978), and Bree- den (1979) – has fallen on hard times.1 Most prominent is the failure to account for the equity premium for any plausible values of risk aversion, which has been referred to as the ‘equity premium puzzle’ (Mehra and Prescott, 1985). These limitations have given rise to a vast literature of promising C-CAPM extensions to achieve a better empirical performance. As a fly in the ointment, Lettau and Ludvigson (2009) show that leading extensions – in- cluding models with long-run risk and recursive preferences (Bansal and Yaron, 2004), habit formation (Campbell and Cochrane, 1999), and limiting participation (Guvenen, 2009) – cannot explain the large and persistent pricing errors found empirically (referred to as Euler equation (EE) errors). Even if this ‘pricing error puzzle’ has not received as much attention as the equity premium puzzle, an open question remains why the leading asset pricing models do fail on that particular dimension (Lettau and Ludvigson, 2009, p.255): “Unlike the equity premium puzzle, these large Euler equation errors cannot be resolved with high values of risk aversion. To explain why the standard model fails, we need to develop [...] models that can rationalize its large pricing errors.” Against this backdrop, our paper makes four key contributions. First, we show that a C-CAPM model with rare disasters in the spirit of Rietz (1988) and Barro (2006, 2009) – i.e. allowing for low-probability events causing infrequent but sharp contractions – not only explains the equity premium (as shown in Barro, 2006), but also rationalizes the large pricing errors found empirically. In fact, the puzzle is not about how to rationalize pricing errors, but rather how to generate empirical pricing errors. Second, we shed light on the source of pricing errors by providing analytical expressions for asset returns, the stochastic discount factor (SDF), and EE errors, both in an endowment economy and in a production economy with low-probability consumption/capital disasters. Our analytical results demonstrate that the EE errors are intimately linked to the poor empirical performance of the C-CAPM found in econometric studies. In particular, we elucidate why the parameter estimates for time preference and risk aversion tend to be severely biased in empirical studies. Third, we run extensive Monte Carlo simulations that demonstrate the impact of low-probability events on the plausibility of standard C-CAPM parameter estimates in small samples when the model is estimated, as is standard, by the generalized method of moments (GMM). We find that 1See Ludvigson (2011) for an excellent survey of the C-CAPM literature. Kroencke (2017) argues that the deficiencies of the classical model might be attributed to a failure to measure consumption correctly. 1 implausibly high estimates for the risk aversion and/or time preference parameters – exactly as found in the empirical literature – naturally arise if market participants expect a future catastrophic change in fundamentals, which just happen not to occur in the sample, or in other words, if the estimation is subject to a so-called peso problem.2 Fourth, to address this issue, we suggest two simple corrections of the moment conditions, both implying more plausible empirical parameter estimates. The novel result of this paper is to show that estimated EE errors may result – despite that GMM’s objective is to minimize pricing errors – together with biased parameter estimates. We present an analytical investigation of EE errors in models with rare events and a data generating process which is able to generate estimated EE errors and biased parameter estimates of similar order of magnitudes as we observe in empirical data. Similar to the statistical approach for heavy-tailed distributions in Kocherlakota (1997), we show that by accounting for rare disasters in the C-CAPM, the model produces reasonable parameter estimates and pricing errors consistent with the empirical data. We show that in small samples where consumption disasters just did not happen to occur, or more generally, where the sample frequency of disasters is not equal to their population frequency, the standard moment conditions are misspecified. This misspecification in turn typically leads to substantial biases of parameter estimates because the objective of GMM is to minimize the squared EE errors. In a simple endowment economy, this objective has two related but unpleasant properties: (i) it essentially removes the pricing errors through (ii) biased parameter estimates of risk aversion and time preference. Only in cases where the minimum of the GMM objective is not sufficiently close to zero, as is often the case in models with changing investment opportunities associated with time-varying interest rates, estimated pricing errors may occur. In line with the results in Lettau and Ludvigson (2009), we illustrate that a model with long-run risk and recursive preferences is unable to generate estimated EE errors, while generating moderately biased estimates of the risk aversion coefficient. Overall, our results thus indicate that rare disaster risk is key for explaining the pricing error puzzle. From a practical standpoint, we put forth several ways of how the biased estimates can be avoided in empirical research by resolving the misspecification in samples with peso problems. Our first proposal starts by assuming that the C-CAPM with rare events is the true data generating process. Then, we use the implied EE errors to remove any misspecification in the moment conditions. While elegant, this remedy is far from perfect in that is not model-free and depends on the assets under consideration. Our second remedy builds on Parker and 2The term ‘peso problem’ is interchangeably with the small-sample inference problems arising from these expected events. The phenomenon is named after events in the Mexican peso market (Lewis, 1992, p.142). 2 Julliard (2005) and includes a set of constants in the moment conditions that are intended to capture any disaster risk without the need of specifying a particular model. Our work relates to the literature on the impact of peso problems on financial markets (cf. Veronesi, 2004).3 While the role of unobserved regime shifts, fat-tailed shocks, and peso problems has been recognized already in earlier literature as a source of misspecification when a C-CAPM is fitted to the data (cf. Kocherlakota, 1997; Saikkonen and Ripatti, 2000), we go beyond the past literature in various ways. First of all, we cast the problem within the rare disaster framework of Rietz (1988) and Barro (2006). In their framework, asset prices reflect risk premia for infrequent and severe disasters in which consumption drops sharply. If disasters are expected by investors ex-ante (reflected in their decisions on consumption and portfolio choice), even if they happen not to occur in sample, a sizeable equity premium can materialize.4 Using historical estimates of consumption disasters for a broad set of countries over a very long period, Barro (2006) shows that a calibrated version of the standard C- CAPM with rare events is able to explain the level of the US equity premium for plausible parameters of risk aversion. We add to this literature by showing that the rare disaster framework helps along two other dimensions: (i) explaining the observed pricing errors that empirical researchers typically encounter in finite sample when fitting a standard C-CAPM, and maybe even more importantly, (ii) explaining the implausible estimates of structural parameters often obtained in empirical work. Our paper also relates to work on the estimation of consumption-based asset pricing mod- els. When taking the C-CAPM to the data, the traditional approach is to estimate the model by GMM. While the advantage of this approach is that it does not rely on a specific structural model, it is sensitive to peso problems. For example, Saikkonen and Ripatti (2000) illustrate the poor performance of the GMM estimator in small and even relatively large samples in the presence of potential regime shifts.